The Ihara zeta function of the infinite grid

TL;DR

This paper computes the Ihara zeta function for the infinite grid, revealing its complex analytic structure, functional equation, and singularities, and connects it to finite grid graphs and tori.

Contribution

It provides the first explicit computation of a non-elementary Ihara zeta function for an infinite graph using elliptic integrals and theta functions.

Findings

Zeta function extends to a multivalued analytic function

Satisfies a functional equation

Finite set of singularities

Abstract

The infinite grid is the Cayley graph of $\mathbb{Z} \times \mathbb{Z}$ with the usual generators. In this paper, the Ihara zeta function for the infinite grid is computed using elliptic integrals and theta functions. The zeta function of the grid extends to an analytic, multivalued function which satisfies a functional equation. The set of singularities in its domain is finite. The grid zeta function is the first computed example which is non-elementary, and which takes infinitely many values at each point of its domain. It is also the limiting value of the normalized sequence of Ihara zeta functions for square grid graphs and torus graphs.